PINN as a PDE Solver
By Prof. Seungchul Lee
http://iai.postech.ac.kr/
Industrial AI Lab at POSTECH
http://iai.postech.ac.kr/
Industrial AI Lab at POSTECH
Table of Contents
1. Lab 3: Euler Beam (Solid Mechanics)¶
1.1. Problem Setup¶
- We will solve a Euler beam problem:
- Problem properties
- Partial differential equations & boundary conditions
$${\partial^4 y \over \partial x^4} + 1 = 0, \qquad \text{where} \quad x \in [0,1]$$
- One Dirichlet boundary condition on the left boundary:
$$y(0) = 0$$
- One Neumann boundary condition on the left boundary:
$$y'(0) = 0$$
- Two boundary conditions on the right boundary:
$$y''(1) = 0, \quad y'''(1) = 0$$
- The exact solution is
$$y(x) = -{1 \over 24}x^4 + {1 \over 6}x^3 - {1 \over 4}x^2$$
- Make a neural network and loss functions like below :
1.2. Solve the Euler Beam problem¶
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!pip install deepxde
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# from deepxde.backend.set_default_backend import set_default_backend
# set_default_backend("tensorflow")
1.2.1. Import Library¶
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import deepxde as dde
import numpy as np
1.2.2. Define PDE System¶
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def dy(x, y):
return dde.grad.jacobian(y, x)
def ddy(x, y):
return dde.grad.hessian(y, x)
def dddy(x, y):
return dde.grad.jacobian(ddy(x, y), x)
def pde(x, y):
dy_xx = ddy(x, y)
dy_xxxx = dde.grad.hessian(dy_xx, x)
return dy_xxxx + 1
1.2.3. Define Boundary Condition¶
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def boundary_left(x, on_boundary):
return on_boundary and np.isclose(x[0], 0)
def boundary_right(x, on_boundary):
return on_boundary and np.isclose(x[0], 1)
1.2.4. Define Geometry, Implement Boundary Condition¶
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geom = dde.geometry.Interval(0, 1)
bc1 = dde.DirichletBC(geom, lambda x: 0, boundary_left) # u(0) = 0
bc2 = dde.OperatorBC(geom, lambda x, y, _: dy(x, y), boundary_left) # u'(0) = 0
bc3 = dde.OperatorBC(geom, lambda x, y, _: ddy(x, y), boundary_right) # u''(1) = 0
bc4 = dde.OperatorBC(geom, lambda x, y, _: dddy(x, y), boundary_right) # u'''(1) = 0
# Reference solution to compute the error
def true_solution(x):
return -(x ** 4) / 24 + x ** 3 / 6 - x ** 2 / 4
data = dde.data.PDE(geom,
pde,
[bc1, bc2, bc3, bc4],
num_domain = 10,
num_boundary = 2,
solution = true_solution,
num_test = 100)
1.2.5. Define Network and Hyper-parameters¶
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layer_size = [1] + [20] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
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model = dde.Model(data, net)
model.compile("adam", lr = 0.001)
1.2.6. Train & Prediction¶
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losshistory, train_state = model.train(epochs = 5000)
dde.saveplot(losshistory, train_state, issave = False, isplot = True)
2. Lab 4: Navier-Stokes Equations (Fluid Mechanics)¶
- Note: strongly recommend you to use GPU rather than CPU. (you can use CoLab GPU for free)
- If you use Google Colab, please implement below codes
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# !pip install deepxde
In [ ]:
import deepxde as dde
import numpy as np
import matplotlib.pyplot as plt
In [ ]:
# from deepxde.backend.set_default_backend import set_default_backend
# set_default_backend("tensorflow")
2.1. Problem Setup¶
- We will solve 2D Navier-Stokes Equations to find velocity profile in infinite parallel plates flow
- Any fluid flowing in plates had to enter at some location. The region of flow near where the fluid enters the plates is termed the entrance region and is illustrated in below figure
- The fluid typically enters the plates with a nearly uniform velocity profile
- As the fluid moves through the plates, viscous effects cause it to stick to the plates wall (no-slip boundary condition)
- Thus, a boundary layer is produced along the plates wall such that the initial velocity profile changes with distance along the plates, $x$, until the fluid reaches the end of the hydrodynamic entrance region (which is also called entrance length) beyond which the velocity profile does not vary with $x$
- Problem properties
$$\rho = 1\operatorname{kg/m^3}, \quad \mu = 1\operatorname{N\cdot s/m^2}, \quad D = 2h = 1\operatorname{m}, \quad
L = 2\operatorname{m}, \quad u_{in} = 1\operatorname{m/s}, \quad \nu = \frac{\mu}{\rho}$$
- Hydraulic diameter is
$$\quad D_h = \lim\limits_{b\to\infty} {4(2bh) \over {2b+4h}} = 4h = 2\operatorname{m}$$
- So, the Reynolds number of this system is
$$Re = \frac{\rho u_{in} D_h}{\mu} = 2 $$
- 2D Navier-Stokes Equations & boundary conditions (for steady state)
$$
\begin{align*}
u{\partial u \over \partial x} + v{\partial u \over \partial y} + {1 \over \rho}{\partial p \over \partial x} - \nu \ \left({\partial^2 u \over {\partial x^2}} + {\partial^2 u \over {\partial y^2}}\right) &= 0\\\\
u{\partial v \over \partial x} + v{\partial v \over \partial y} + {1 \over \rho}{\partial p \over \partial y} - \nu \ \left({\partial^2 v \over {\partial x^2}} + {\partial^2 v \over {\partial y^2}}\right) &= 0\\\\
{\partial u \over \partial x} + {\partial v \over \partial y} &= 0
\end{align*}
$$
- Two Dirichlet boundary conditions on the plate boundary (no-slip condition),
$$u(x,y) = 0, \quad v(x,y) = 0 \qquad \text{at} \quad y = \frac{D}{2} \ \; \text{or} \; -\frac{D}{2}$$
- Two Dirichlet boundary conditions at the inlet boundary
$$u(-1,y) = u_{\text{in}}, \quad v(-1,y) = 0$$
- Two Dirichlet boundary conditions at the outlet boundary
$$p(1,y) = 0, \quad v(1,y) = 0$$
- Make a neural network and loss functions like below :
2.2. CFD Solution¶
- CFD solution of this problem is illustrated in below figures
- Velocity u and velocity v, and pressure p, respectively
- Solve this problem using PINN and then compare with CFD solutions
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# Properties
rho = 1
mu = 1
u_in = 1
D = 1
L = 2
2.3.2. Define PDE with Boundary & Initial Conditions¶
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def boundary_wall(X, on_boundary):
on_wall = np.logical_and(np.logical_or(np.isclose(X[1], -D/2), np.isclose(X[1], D/2)), on_boundary)
return on_wall
def boundary_inlet(X, on_boundary):
return on_boundary and np.isclose(X[0], -L/2)
def boundary_outlet(X, on_boundary):
return on_boundary and np.isclose(X[0], L/2)
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def pde(X, Y):
du_x = dde.grad.jacobian(Y, X, i = 0, j = 0)
du_y = dde.grad.jacobian(Y, X, i = 0, j = 1)
dv_x = dde.grad.jacobian(Y, X, i = 1, j = 0)
dv_y = dde.grad.jacobian(Y, X, i = 1, j = 1)
dp_x = dde.grad.jacobian(Y, X, i = 2, j = 0)
dp_y = dde.grad.jacobian(Y, X, i = 2, j = 1)
du_xx = dde.grad.hessian(Y, X, i = 0, j = 0, component = 0)
du_yy = dde.grad.hessian(Y, X, i = 1, j = 1, component = 0)
dv_xx = dde.grad.hessian(Y, X, i = 0, j = 0, component = 1)
dv_yy = dde.grad.hessian(Y, X, i = 1, j = 1, component = 1)
pde_u = Y[:,0:1]*du_x + Y[:,1:2]*du_y + 1/rho * dp_x - (mu/rho)*(du_xx + du_yy)
pde_v = Y[:,0:1]*dv_x + Y[:,1:2]*dv_y + 1/rho * dp_y - (mu/rho)*(dv_xx + dv_yy)
pde_cont = du_x + dv_y
return [pde_u, pde_v, pde_cont]
2.3.3. Define Geometry, Implement Boundary Condition¶
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geom = dde.geometry.Rectangle(xmin=[-L/2, -D/2], xmax=[L/2, D/2])
bc_wall_u = dde.DirichletBC(geom, lambda X: 0., boundary_wall, component = 0)
bc_wall_v = dde.DirichletBC(geom, lambda X: 0., boundary_wall, component = 1)
bc_inlet_u = dde.DirichletBC(geom, lambda X: u_in, boundary_inlet, component = 0)
bc_inlet_v = dde.DirichletBC(geom, lambda X: 0., boundary_inlet, component = 1)
bc_outlet_p = dde.DirichletBC(geom, lambda X: 0., boundary_outlet, component = 2)
bc_outlet_v = dde.DirichletBC(geom, lambda X: 0., boundary_outlet, component = 1)
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data = dde.data.PDE(geom,
pde,
[bc_wall_u, bc_wall_v, bc_inlet_u, bc_inlet_v, bc_outlet_p, bc_outlet_v],
num_domain = 3000,
num_boundary = 500,
num_test = 1000,
train_distribution = 'LHS' )
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plt.figure(figsize = (10,8))
plt.scatter(data.train_x_all[:,0], data.train_x_all[:,1], s = 0.5)
plt.xlabel('x-direction length')
plt.ylabel('Distance from the middle of plates (m)')
plt.show()
2.3.4. Define Network and Hyper-parameters¶
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layer_size = [2] + [64] * 5 + [3]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
model = dde.Model(data, net)
model.compile("adam", lr = 1e-3)
2.3.5. Train (Adam Optimizer)¶
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losshistory, train_state = model.train(epochs = 10000)
dde.saveplot(losshistory, train_state, issave = False, isplot = True)
2.3.6. Plot Results (Adam Optimizer)¶
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samples = geom.random_points(500000)
result = model.predict(samples)
color_legend = [[0, 1.5], [-0.3, 0.3], [0, 35]]
for idx in range(3):
plt.figure(figsize = (20, 4))
plt.scatter(samples[:, 0],
samples[:, 1],
c = result[:, idx],
cmap = 'jet',
s = 2)
plt.colorbar()
plt.clim(color_legend[idx])
plt.xlim((0-L/2, L-L/2))
plt.ylim((0-D/2, D-D/2))
plt.tight_layout()
plt.show()
2.3.7. Train More (L-BFGS Optimizer)¶
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dde.optimizers.config.set_LBFGS_options(maxiter=3000)
model.compile("L-BFGS")
losshistory, train_state = model.train()
dde.saveplot(losshistory, train_state, issave = False, isplot = True)
2.3.8. Plot Results (Adam + L-BFGS)¶
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samples = geom.random_points(500000)
result = model.predict(samples)
color_legend = [[0, 1.5], [-0.3, 0.3], [0, 35]]
for idx in range(3):
plt.figure(figsize = (20, 4))
plt.scatter(samples[:, 0],
samples[:, 1],
c = result[:, idx],
cmap = 'jet',
s = 2)
plt.colorbar()
plt.clim(color_legend[idx])
plt.xlim((0-L/2, L-L/2))
plt.ylim((0-D/2, D-D/2))
plt.tight_layout()
plt.show()
2.3.9. Validation: Plot Velocity Profile at the End of the Plate¶
- Fully developed velocity profile at the infinite parallel plates flow are known as
$$u(y) = {3V_{avg} \over 2} \left[ 1- \left( \frac{y}{h} \right)^2 \right]$$
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# Analytic solution
x = np.ones([1000,1])
y = np.linspace(-0.5, 0.5, 1000).reshape(1000,1)
outlet = np.hstack([x, y])
analytic_solution = u_in * 1.5 * (1 - ((y)/(D/2))**2)
PINN_solution = model.predict(outlet)
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plt.figure(figsize = (10,8))
plt.plot(y, PINN_solution[:, 0], c = 'tomato', linewidth = 3, label = 'PINN solution')
plt.plot(y, analytic_solution, c = 'k', linestyle = '--', label = 'Analytic solution')
plt.xticks(np.linspace(-0.5, 0.5, 5), fontsize = 15)
plt.yticks(np.linspace(0, 1.5, 11), fontsize = 15)
plt.legend(fontsize = 12)
plt.xlabel('Distance from the middle of plates (m)', fontsize = 15)
plt.ylabel('Velocity ($u$)', fontsize = 15)
plt.show()
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